WRC RESEARCH REPORT NO. 21 SOME DATA ON DIFFUSION AND TURBULENCE IN RELATION TO REAERATION E.R. Holley, Jr. Department of Civil Engineering Uviversity of Illinois, Urbana FINAL REPORT Project No.A -021-11-L September 1, 1967 - June 30, 1968 The work upon which this publication is based was supported by funds provided by the U.S. Department of the Interior as authorized under the Water Resources Research Act o f 1964, P.L. 88-379 Agreement No. 14-01-0001-1081 UNIVERSITY OF ILLINOIS WATER RESOURCES CENTER 3220 Civi 1 Engineering Building Urbana, I l l i n o i s 61801 July 1969
Transcript
WRC RESEARCH REPORT NO. 21
SOME DATA ON DIFFUSION AND TURBULENCE IN RELATION TO REAERATION
E . R . H o l l e y , J r . Department o f C i v i l E n g i n e e r i n g U v i v e r s i t y o f I l l i n o i s , Urbana
F I N A L R E P O R T
P r o j e c t No.A -021-11-L
September 1 , 1967 - June 30, 1968
The work upon wh ich t h i s p u b l i c a t i o n i s based was suppor ted by funds p r o v i d e d by t h e U.S. Department o f t h e I n t e r i o r as a u t h o r i z e d under
t h e Water Resources Research Ac t o f 1964, P .L . 88-379 Agreement No. 14-01-0001-1081
UNIVERSITY OF ILLINOIS WATER RESOURCES CENTER
3220 C i v i 1 E n g i n e e r i n g B u i l d i n g Urbana, I l l i n o i s 61801
J u l y 1969
ABSTRACT
SOME DATA ON DIFFUSION AND TURBULENCE I N RELATION TO REAERATION
Heat was used as a t r a c e r t o determine d i f f u s i o n r a t e s immediately below t h e f r e e su r face bo th i n t he " f i l m " r eg i on and i n t h e remainder o f the water i n a m i x i ng vessel . The r e s u l t s t end t o i n d i c a t e t h a t a d i f f u s i o n model can be used t o represent t h e downward t r a n s p o r t o f a substance which i s be ing absorbed a t the f r e e su r face . Apparen t l y , t he d i f f u s i o n c o e f f i c i e n t i n t he " f i l m " can be e i t h e r equal t o o r g r e a t e r than t he mo lecu la r c o e f f i c i e n t depending on t he amount o f m i x i ng a t t he f r e e sur- face. Hot f i l m anemometry was used t o determine tu rbu lence c h a r a c t e r i s - t i c s f rom 1 i n . t o 0.006 i n . below the f r e e su r f ace o f a l a b o r a t o r y open channel f l ow . The energy spec t ra i n d i c a t e no s i g n i f i c a n t changes i n t he tu rbu lence i n t h i s reg ion . Thus, appa ren t l y tu rbu lence e x i s t s r i g h t up t o t he f r e e su r f ace and i n t he " f i l m " reg ion . The amount o f r e l i a b l e data t h a t was ob ta ined was very l i m i t e d .
Ho l l ey , E . R . SOME DATA ON DIFFUSION AND TURBULENCE I N RELATION TO REAERATION Research Report No. 21, Water Resources Center, U n i v e r s i t y o f I l l i n o i s , Ju l y , 1969, Urbana, I l l i n o i s
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KEYWORDS--"reaera t ion/"'di sso lved oxygen/di f f u s i on / t u rbu lence
Domestic and i n d u s t r i a l wastes a r e commonly d ischarged i n t o
r i v e r s and e s t u a r i e s . A m a j o r problem i s the de te rm ina t i on o f the
amount and concen t ra t i on o f waste m a t e r i a l s which can be ass imu la ted
by a g i ven r i v e r . Th i s depends t o a l a r g e e x t e n t on t he amount o f
d i s s o l v e d oxygen which i s p resen t i n t he waterway. I n o rde r t o de te r -
mine how much oxygen w i l l be p resen t under a g i ven s e t o f c o n d i t i o n s ,
i t i s necessary t o determine t he va r i ous ways i n which oxygen i s
supp l i ed and u t i l i z e d . Furthermore, i t i s necessary t o know the r a t e s
a t which the processes t ake p l ace and the f a c t o r s which i n f l u e n c e
these ra tes . One s i g n i f i c a n t source o f d i s so l ved oxygen i n a r i v e r
i s su r face reae ra t i on , i .e. the abso rp t i on o f oxygen from the atmosphere.
Under a g i ven s e t o f c o n d i t i o n s , t h e r e e x i s t s a concen t ra t i on
o f d i s so l ved oxygen f o r which the s o l u t i o n o f oxygen i n water w i l l be
i n e q u i l i b r i u m w i t h the atmosphere t o which t he water i s exposed. Th i s
c o n d i t i o n i s c a l l e d s a t u r a t i o n and t he cor responding concen t ra t i on i s
g i ven t he symbol c . I f t h i s e q u i l i b r i u m c o n d i t i o n i s d i s t u rbed , the S
concen t ra t i on ( c ) tends t o r e t u r n t o t he e q u i l i b r i u m c o n d i t i o n . For
example, i f r i v e r water i s sa tu ra ted w i t h oxygen and then substances
posessing b iochemica l oxygen demand (BOD) a re in t roduced i n t o t h e r i v e r ,
the BOD w i l l r e a c t w i t h t he d i sso l ved oxygen, caus ing a r educ t i on i n t he
oxygen con ten t . Then, when C i s l e s s than c oxygen i s absorbed from s '
the atmosphere. The r a t e a t which t h e abso rp t i on takes p l ace depends
on many f a c t o r s among which a r e temperature, type and d i s t r i b u t i o n o f
oxygen demanding substances, t he presence o f su r f ace -ac t i ve agents, and
t he tu rbu lence l e v e l i n t h e wa te r . Th i s r e p o r t dea ls p r i m a r i l y w i t h the
r e l a t i o n between r e a e r a t i o n and t h e t u r b u l e n c e l e v e l and r e l a t e d
t u r b u l e n c e parameters .
I n t u r b u l e n t wa te r , t h e r a t e o f a b s o r p t i o n o f oxygen i s -9-
assumed t o be p r o p o r t i o n a l t o t h e oxygen d e f i c i t D so t h a t ( 2 3 ) "
where D = c - C and K may be c a l l e d a r e a e r a t i o n c o e f f i c i e n t . I f s 2
t h e i n i t i a l d e f i c i t a t t = 0 i s D t hen E q . 1-1 may be i n t e g r a t e d 0 ,
t o g i v e
where k = K2/2.30. 2
For many yea rs , a t t e m p t s have been made t o f i n d c o n s i s t e n t l y
v a l i d c o r r e l a t i o n s between r e a e r a t i o n c o e f f i c i e n t s such a s k and 2
h y d r a u l i c parameters . Some o f t h e c o r r e l a t i o n s t h a t have been o b t a i n e d
a r e rev iewed i n t h e n e x t s e c t i o n o f t h i s r e p o r t . As y e t , t h e r e appears
t o be no r e l a t i o n s h i p wh ich has r e c e i v e d w ide acceptance among t h e v a r i -
ous persons d e a l i n g w i t h r e a e r a t i o n prob lems. The a u t h o r f e e l s t h a t
t h i s l a c k o f a g e n e r a l l y v a l i d r e l a t i o n s h i p r e s u l t s f rom t h e f a c t t h a t
.*- Numbers i n p a r e n t h e s i s r e f e r t o r e f e r e n c e s l i s t e d a t the end o f t h e r e p o r t .
t h e m a j o r i t y o f t h e c o r r e l a t i o n s have been ob ta i ned p r i m a r i l y by data-
f i t t i n g techniques. Resu l t s f rom t h i s t ype o f approach w i l l a lways
r e f l e c t t h e p a r t i c u l a r c h a r a c t e r i s t i c s o f t he s i t u a t i o n i n which they
were eva luated. Th i s i s t r u e even though a t tempts a r e made t o i n c l u d e
t h e e f f e c t s o f tu rbu lence . Probably, a g e n e r a l l y a p p l i c a b l e express ion
f o r r e a e r a t i o n c o e f f i c i e n t s w i l l n o t r e s u l t u n t i l t h i s express ion can
be based on an unders tanding o f t h e mechanism by which t u rbu lence
a f f e c t s t h e r a t e o f oxygen a b s o r p t i o n r a t h e r than be ing based on p u r e l y
e m p i r i c a l c o r r e l a t i o n s . I t i s hoped t h a t t h e research desc r i bed i n
t h i s r e p o r t i s one s tep toward ach iev i ng an unders tanding o f t h e mech-
anisms i nvo l ved i n r e a e r a t i o n and t he r e l a t i o n o f these mechanisms t o
t he h y d r a u l i c s o f t he f l ow .
However, u n t i l t h e r e a e r a t i o n mechanisms a r e f u l l y understood,
i t i s s t i l l necessary t o ana lyze p o l l u t i o n problems i n r i v e r s . Thus,
a d d i t i o n a l e m p i r i c a l c o r r e l a t i o n s f o r r e a e r a t i o n c o e f f i c i e n t s a r e s t i l l
appear ing i n t h e 1 i t e r a t u r e and a r e f i n d i n g use i n r e l a t i n g r e a e r a t i o n
c o e f f i c i e n t s t o mean h y d r a u l i c parameters o f r i v e r s and e s t u a r i e s .
2) D I S C U S S I O N OF P R E V I O U S WORK
2.1) General Empi r i c a l Approaches
The work o f S t r e e t e r and Phelps (23) i n 1925 i s g e n e r a l l y
cons idered t o have marked t he beg inn ing o f a new era i n t h e a n a l y s i s
o f stream p o l l u t i o n problems. P r i o r t o t h e i r work on t h e Ohio River ,
t h e a l l o w a b l e p o l l u t i o n l o a d f o r a g i v e n stream was determined p r i -
mar i l y by a " r u l e o f thumb" d i l u t i o n r a t i o between sewage d ischarge
and stream f low. S t r e e t e r and Phelps t ook a more r a t i o n a l approach
and presented an express ion which e f f e c t i v e l y represented t he oxygen
ba lance o f a stream under a g i ven se t o f assumptions. S t r e e t e r and
Phelps used t h e equa t ion
where D i s t h e oxygen d e f i c i t ( c - c ) , C i s t h e average concen t ra t i on S
o f oxygen a t a sec t ion , c i s t he s a t u r a t i o n concen t ra t i on , K1 i s t h e S
deoxygenat ion ra te , L i s t h e amount o f oxygen demand i n t h e water, K2
i s t he r e a e r a t i o n r a te , and t i s t ime. Eq. 2-1 represen ts t h e r a t e o f
change o f oxygen d e f i c i t due t o u t i 1 i z a t i on o f oxygen ( i .e. deoxygena-
t i o n ) by t h e p o l l u t a n t and due t o su r f ace a e r a t i o n , which was assumed
t o be t h e o n l y source f o r a d d i t i o n a l d i s s o l v e d oxygen. T h i s expres-
s i o n was taken as a p p l y i n g t o a one-dimensional element o f water as
i t moved down t h e r i v e r , and t was i n t e r p r e t e d as t h e t ime o f f l o w
(x /u ) . Thus, Eq. 2-1 was e f f e c t i v e l y assumed t o be
where U i s the mean v e l o c i t y (which may vary w i t h x) and x i s t h e
l o n g i t u d i n a l coo rd i na te a l ong t h e stream.
I n t e g r a t i o n o f Eq. 2-2 leads t o t h e oxygen-sag curve.
S t r e e t e r and Phelps made a l a r g e number o f measurements o f oxygen
concen t ra t i o n t o descr ibe t h i s oxygen- sag curve i n v a r i o u s reaches o f
t he Ohio R ive r a t va r i ous t imes d u r i n g the year . By t a k i n g va lues o f
K determined i n the l a b o r a t o r y , they were a b l e t o c a l c u l a t e r e a e r a t i o n 1
r a t e s f rom t h e i r data . I t was argued t h a t K2 should be i n v e r s e l y pro-
p o r t i o n a l t o t he square o f the depth o f t h e r i v e r and d i r e c t l y propor-
t i o n a l t o t he t u rbu lence l e v e l o f t he r i v e r , f rom which i t was p o s t u l a t e d
t h a t
where m and n were assumed t o be cons tan ts f o r va r i ous s t r e t c h e s o f t he
r i v e r and H was taken as t he depth above extreme low water . ( ~ o t e t h a t
, m i s n o t dimensionless.) Values o f m and r i were e m p i r i c a l l y determined
f o r the va r i ous reaches o f t he Ohio R i ve r . Thus, any inaccurac ies
p resen t i n t he assumption t h a t K i s i n v e r s e l y p r o p o r t i o n a l t o H2 s imply 2
i n f l uenced t h e e m p i r i c a l va l ue o f m and n r a t h e r than render ing Eq. 2-3
i n v a l i d . I n e v a l u a t i n g K S t r e e t e r and Phelps assumed t h a t su r f ace 2'
r eae ra t i on was t h e o n l y source o f oxygen supply .
The work done by C h u r c h i l l e t a1 a t TVA (2) i s another emp i r i -
c a l approach. F i e l d measurements were made i n r i v e r s which were s a i d t o
have e s s e n t i a l l y no oxygen demand and no source o f oxygen o t h e r than
su r f ace absorp t ion . The r i v e r reaches were se lec ted so as t o be
r e l a t i v e l y un i fo rm. From a t o t a l o f 30 exper iments i n 5 r i v e r s , i t
was concluded t h a t k2 a t 2 0 ' ~ c o u l d be adequate ly represented by
where k = K2/2.30 (.day-'), U i s t h e average f l ow v e l o c i t y ( f t / s e c ) , 2
and h i s t h e h y d r a u l i c mean depth ( f t ) . I n a d i scuss ion o f t he work
o f C h u r c h i l l e t a l , H u l l and de F i l i p p i (12) p o i n t e d o u t t h a t Eq. 2-4
was compared o n l y w i t h t he data used t o eva lua te t h e e m p i r i c a l con-
s t a n t s and t h a t no independent v e r i f i c a t i o n o f Eq. 2-4 had y e t been
presented.
Changes i n tu rbu lence c o n d i t i o n s i n a r i v e r , r a t h e r than
<,
changes i n mean f l ow parameters pe r se, account f o r changes i n t he re-
a e r a t i o n r a t e k Eq. 2-4 g i ves a c o r r e l a t i o n between k and two mean 2 ' 2
f l o w parameters U and h, no t between k and t u rbu lence parameters. I f 2
t h e r e were a un ique r e l a t i o n between tu rbu lence c o n d i t i o n s and U and
h, then Eq. 2-4 would e f f e c t i v e l y r e l a t e k t o the t u rbu lence cond i t i ons , 2
bu t t h i s i s n o t necessa r i l y t h e case. C h u r c h i l l e t a1 p o i n t ou t t h a t
e i t h e r t h e channel s lope (s) o r t h e Darcy-Weisbach f r i c t i o n f a c t o r ( f )
i s needed i n a d d i t i o n t o U and h i n o rde r t o s p e c i f y t he mean turbu-
l ence c o n d i t i o n , b u t they go on t o p o i n t ou t t h a t i n t h e i r work S was
h i g h l y c o r r e l a t e d w i t h bo th U and h. Th i s seems t o be equ i va l en t t o
say ing t h a t a l l t h e streams had e s s e n t i a l l y the same h y d r a u l i c roughness.
Thus, no improvement i n t h e c o r r e l a t i o n between k and f l o w parameters 2
was ach ieved by the i n c l u s i o n o f S i n t he c o r r e l a t i o n . I t should be
p o i n t e d ou t t h a t a l l t h e data used i n o b t a i n i n g Eq. 2-4 came f rom r i v e r s
i n one s e c t i o n o f t he coun t r y . I n o t h e r reg ions where streams have
d i f f e r e n t c h a r a c t e r i s t i c s f rom those used by C h u r c h i l l e t a l , a d i f -
f e r e n t c o r r e l a t i o n between S, U, and h may e x i s t and cause Eq. 2-4
t o be i naccu ra te f o r o the r reg ions .
Krenke l (13) and Krenkel and Or lob (14) performed l a b o r a t o r y
exper iments on t he r a t e o f r e a e r a t i o n i n t u r b u l e n t shear f l ow. They
t ook t h e c o e f f i c i e n t o f l o n g i t u d i n a l mass t r a n s p o r t ( i . e . t he coef-
f i c i e n t o f l o n g i t u d i n a l d i spe rs i on , E) as be ing c h a r a c t e r i s t i c o f
t h e genera l s t a t e o f t u r b u l e n t m i x i ng i n t h e f low, and reasoned t h a t
t h i s c o e f f i c i e n t cou ld be used t o represen t t h e e f f e c t s o f tu rbu lence
on k2. A good c o r r e l a t i o n between k and E was found f o r t h e i r data . 2
I n a d i scuss ion o f K renke l and O r l o b ' s paper, Harleman and Ho l l ey (9)
p o i n t e d o u t t h a t t h e r a t e o f l o n g i t u d i n a l mass t r a n s p o r t ( i .e . d i s -
pe r s i on ) which Krenkel and Or l ob used r e s u l t s p r i m a r i l y f rom a
convec t i ve mechanism r a t h e r than be ing a d i r e c t r e s u l t o f tu rbu lence .
Thus, i t does no t seem l o g i c a l t o take t h e c o e f f i c i e n t o f l o n g i t u d i n a l
d i s p e r s i o n as r e p r e s e n t a t i v e o f tu rbu lence e f f e c t s . Harleman and
Ho l l ey f u r t h e r i n d i c a t e d t h a t l o n g i t u d i n a l d i s p e r s i o n u s u a l l y p l a y s
no r o l e i n t he oxygen abso rp t i on process i n r i v e r s . However, because
o f the c h a r a c t e r i s t i c s o f t u r b u l e n t shear f l ow , l o n g i t u d i n a l d i spe rs i on
(a convec t i ve process) i s h i g h l y c o r r e l a t e d w i t h t h e average v e r t i c a l
eddy d i f f u s i v i t y (a t u r b u l e n t d i f f u s i o n p rocess) . Thus, the apparent
c o r r e l a t i o n found between k2 and E can more l o g i c a l l y be thought o f as
a c o r r e l a t i o n between k and the average v e r t i c a l eddy d i f f u s i v i t y , 2
which i s i n d i c a t i v e o f t he c h a r a c t e r i s t i c s o f t h e tu rbu lence i n t h e
f l ow. I t has more r e c e n t l y been p o i n t e d o u t by F ischer (7) t h a t
l o n g i t u d i n a l d i spe rs i on may depend s t r o n g l y on t r ansve rse v e l o c i t y
d i s t r i b u t i o n s . Th i s i s ano ther reason t h a t t h e d i s p e r s i o n c o e f f i c i e n t
i s p robab ly n o t a good parameter f o r c o r r e l a t i o n w i t h r e a e r a t i o n r a t e
c o e f f i c i e n t s . Krenkel and Or lob a l s o showed t h a t k f o r t h e i r data 2
was r e l a t e d t o t h e r a t e o f t u r b u l e n t energy d i s s i p a t i o n f o r t h e f l ow.
The most recen t e m p i r i c a l approach was repo r t ed by Thackston
and Krenkel (24) . They cons idered t he genera 1 dependence o f reaera t ion
on t u r b u l e n t d i f f u s i o n i n o rde r t o o b t a i n t h e h y d r a u l i c parameters t o
which k was c o r r e l a t e d . From bo th l a b o r a t o r y and f i e l d data, they 2
ob ta i ned t he express ion
where g i s t h e g r a v i t a t i o n a l acce le ra t i on , and they i n d i c a t e d t h a t t h i s
equa t ion gave a b e t t e r o v e r a l l f i t t o t h e data than t he o t h e r g e n e r a l l y
-used equat ions. Never the less, t h e r e was s t i l l cons ide rab le s c a t t e r i n
t h e data w i t h respec t t o a l i n e r ep resen t i ng Eq. 2-5 and t h e r e was one
s e t o f data which c o n s i s t e n t l y gave k va lues about t w i c e as g r e a t as 2
would be i n d i c a t e d by Eq. 2-5.
2.2) F i l m Theory
F i l m theory represen ts one o f t h e a n a l y t i c a l approaches t h a t
has been used t o study t h e a b s o r p t i o n o f gases i n t o t u r b u l e n t water .
T h i s concept was presented by Whitman (25) i n 1923 and by Lewis and
Whitman (16) i n 1924. As a p p l i e d t o t h e a b s o r p t i o n o f oxygen by
water , t h i s theory s t a t e s t h a t t he r a t e o f abso rp t i on i s c o n t r o l l e d
by t h e r a t e o f d i f f u s i o n through a s tagnant water f i l m a t t h e water
su r face . I t i s f u r t h e r argued t h a t t he d i s s o l v e d oxygen a t t h e sur-
face i s i n e q u i l i b r i u m w i t h t he atmosphere,and thus t h e c o n c e n t r a t i o n
a t t h e i n t e r f a c e i s t h e s a t u r a t i o n concen t ra t i on , c . Turbulence i s S
assumed t o keep t he water w e l l enough mixed below t h e f i l m so t h a t
t h e concen t ra t i on i s un i f o rm everywhere except i n the f i l m . Thus,
through t h e t h i n f i l m , a steep c o n c e n t r a t i o n g r a d i e n t e x i s t s , as
shown i n F ig . 2-1.
water su r f ace c = c S
su r f ace f i l m - - - - .
F i g . 2-1: Concen t ra t ion d i s t r i b u t i o n assumed by f i l m theory .
The d i f f u s i o n through t h e f i l m i s s a i d t o be s o l e l y by mo lecu la r a c t i o n
s i n c e t he f i l m i s assumed t o be s tagnant . Thus, d i f f u s i o n through t h e
f i l m f o l l o w s F i c k ' s law (1) which s t a t e s t h a t t he r a t e o f mass t r a n s p o r t (q)
through an area (A) i s equal t o a mo lecu la r d i f f u s i o n c o e f f i c i e n t (D,)
t imes t he n e g a t i v e concen t ra t i on g r a d i e n t pe rpend i cu l a r t o A. A p p l i e d
t o a t r a n s p o r t i n t h e v e r t i c a l (y ) d i r e c t i o n , F i c k ' s law i s
where A i s t he su r f ace area. The c o n c e n t r a t i o n d i s t r i b u t i o n i s assumed S
t o be l i n e a r through t h e f i l m so t h a t t h e f i l m o f th i ckness b,
where C i s t he concen t ra t i on below t he f i l m . F i c k ' s law f o r t he f i l m
then becomes
The minus s i g n i s sometimes o m i t t e d w i t h the unders tanding t h a t t he
t r a n s p o r t i s downward when C i s l e s s than c . S
F i l m theory a l s o l e d t o t he d e f i n i t i o n o f a l i q u i d f i l m co-
e f f i c i e n t ( K ~ ) by t he express ion
= A K ( c - C ) = A K D qy S L S s L
Thus, K i s t h e r a t e o f downward d i f f u s i o n o f oxygen ( i . e . q ) per u n i t L Y
su r f ace area ( A ~ ) per u n i t oxygen d e f i c i t (D) . Under t h e assumptions
o f f i l m theory ,
No method was presented by Lewis and Whitman f o r f i n d i n g b under genera l
c o n d i t i o n s . Thus, even i f Dm i s known f o r a g i v e n se t o f c o n d i t i o n s and
i f t h e assumptions o f f i l m theory a r e v a l i d , t h i s r e p r e s e n t a t i o n i s n o t
very h e l p f u l f o r p r e d i c t i n g r eae ra t i o n r a tes .
2.3) Sur face Renewal a n d Re la ted Approaches
Ob jec t i ons were r a i s e d t o t h e b a s i c f i l m t heo ry o f Lewis and
Whitman because i t assumed t h e f i l m t o be s tagnan t . As an a l t e r n a t i v e ,
H i g b i e (10) proposed t h e p e n e t r a t i o n theory . T h i s theory f o r t h e r a t e
o f gas abso rp t i on was based on t h e assumption t h a t t h e whole body o f
water was s tagnant f o r s h o r t p e r i o d s o f t ime and, d u r i n g these per iods ,
oxygen was absorbed and d i f f u s e d downward s o l e l y by mo lecu la r d i f f u s i o n .
Then, p e r i o d i c a l l y , t h e wa te r was ins tan taneous ly and comple te ly mixed.
T h i s model l e t t o an equa t ion s imu la r t o Eq. 2-8 w i t h KL g i v e n by
where t 1 i s t h e average t ime between t h e complete m ix ings .
Danckwerts (3 ) extended Higb i e ' s approach by assuming t h a t
va r i ous v e r t i c a l elements o f t he water c o u l d i n d i v i d u a l l y undergo com-
p l e t e v e r t i c a l m i x i ng w i t h d i f f e r e n t p e r i o d s between m i x i n g ~ . He
assumed t h a t t he s t a t i s t i c a l d i s t r i b u t i o n o f t h e m i x i ng was descr ibed
by
where f ( t ) i s t h e p r o p o r t i o n a l p a r t o f t he v e r t i c a l e lements f o r which
t h e e lapsed t ime s i n c e t h e l a s t m i x i n g i s between t and t + d t . The
cons tan t r may be i n t e r p r e t e d as t h e average r a t e a t which v e r t i c a l
m i x i ng takes p lace . Thus, r i s analogous t o l / t l o f E q . 2-10.
Danckwertsl approach aga in l e d t o E q . 2-7 f o r .the r a t e o f a b s o r p t i o n
where K was g i ven by L
Dobbins (4, 5) used t he concept o f a su r f ace f i l m , b u t pos-
t u l a t e d t h a t t h e water i n t h e f i l m was p e r i o d i c a l l y mixed w i t h water
from below t he f i l m . I n a sense, t h i s was a combinat ion o f f i l m theory . and Danckwerts' work. I n Dobbins1 model, t h e r eg ion below t h e f i l m
was assumed t o be u n i f o r m l y mixed a t a l l t imes w h i l e new su r f ace f i l m
was c o n t i n u a l l y be ing c rea ted and then removed. I t was assumed t h a t
o n l y modecular d i f f u s i o n takes p l a c e w h i l e a g i ven p a r c e l o f water i s
i n t h e f i l m . Dobbins a l s o assumed t h a t v a r i o u s p a r t s o f t h e f i l m were
mixed o r "renewed" a t a r a t e such t h a t E q . 2-11 descr ibed t h e d i s t r i b u -
t i o n o f t imes ( o r ages) s i n c e t h e l a s t renewal w i t h i n t he f i l m . For a
f i l m o f t h i c kness b, he showed (4) t h a t these assumptions l e a d t o
As r -+ 0, KL o f E q . 2-13 approaches KL o f E q . 2-9, w h i l e f o r l a r g e r,
E q . 2-13 approaches E q . 2-12.
I n r e f . 5, Dobbins d e s c r i b e s some exper imen ts wh ich were
conducted t o i n v e s t i g a t e t h e v a l i d i t y o f E q . 2-13. T e s t s were made
on t h e r a t e s o f a b s o r p t i o n o f h e l i u m and n i t r o g e n i n t o w a t e r wh ich
was b e i n g mixed by a v e r t i c a l l y o s c i l l a t i n g s t a c k o f m e t a l g r i d s . By
measur ing v a l u e s o f K f o r t h e two gases, i t was p o s s i b l e t o c a l c u l a t e L
v a l u e s o f r and b. A c o n s i s t a n t c o r r e l a t i o n was found wh ich r e l a t e d
t h e v a l u e s o f r and b t o t h e parameters d e s c r i b i n g t h e o s c i l l a t i o n o f
t h e m e t a l g r i d s . As wou ld be expected, r i nc reased and b decreased
a s t h e speed o f o s c i l l a t i o n (a) o f t h e g r i d s was increased. However,
t h e changes wh ich were found f o r r a r e somewhat d i f f i c u l t t o under-
s tand. R e c a l l t h a t r was assumed t o be t h e average r a t e a t wh ich
t h e s u r f a c e f i l m i s renewed. For a a o f 18 rpm (113 rad/min) , r was
c a l c u l a t e d t o be 0.171 p e r min, w h i l e f o r a = 196 rpm (1230 rad/min) ,
r was 3780 p e r m i r ! f o r t h e same a m p l i t u d e o f screen mot ion . I t i s
d i f f i c u l t t o conce ive t h a t i n c r e a s i n g o by a f a c t o r o f abou t 11 wou ld
have changed t h e c h a r a c t e r i s t i c s o f t h e m i x i n g t o such an e x t e n t t h a t
r wou ld be i nc reased by a f a c t o r g r e a t e r than 20,000.
O'Connor and Dobbins (20) a p p l i e d E q . 2-13 t o r e a e r a t i o n i n
n a t u r a l s t reams. From t y p i c a l v a l u e s f o r r, b, and Dm, they argued
t h a t t h e "co th" te rm i n E q . 2-13 s h o u l d be e s s e n t i a l l y u n i t y For n a t u r a l
s t reams. Then, K wou ld be L
T h i s i s i d e n t i c a l t o Danckwer ts ' e x p r e s s i o n ( E ~ . 2-12). Streams were
d i v i d e d i n t o t h ~ s e possess ing ! ! n o n i s o t r o p i c " t u r b u l e n c e and those wh ich
were assumed to posses "isotropic" turbulence. The distribution
between the types of turbulence was based on the Chezy coefficient
C . If C was less than 17, nonisotropy was assumed, while C greater C C C
than 17 was taken as implying isotropy. By using this relative low
critical value of C the majority of the streams which they analyzed c ,
were classified as having isotropic turbulence. Note, however, that
the concept of isotropic turbulence in shear flow (such as a river)
is contrary to the basic mechanics of turbulence (21, Chap. 6).
O'Connor and Dobbins argued that r should be given by the
absolute value of the average turbulent veloci ty fluctuation divided
by the turbulent mixing length, where both quantities are evaluated
at the free surface. This led to the conclusion, for nonisotropic
turbulence, that r equals the velocity gradient at the free surface.
Using the logarithmic velocity distribution to evaluate this gradient
for two dimensional flow, they obtained
and
where u = .von Karman's constant (taken as 0.4), k = reaeration coef- 2 - 1
ficient (day ) , and ft-sec units are used on the right hand side
of Eq. 2-16. (Note that the logarithmic velocity distribution has
been found to give poor agreement with the velocities near the free
sur face. See r e f . 21, p . 489 and 518.) For i s o t r o p i c t u rbu lence
( i . e . Cc g r e a t e r than 17), r was s a i d t o be
l ead ing t o
where U i s t he average v e l o c i t y ( f t / s e c ) and t he o t h e r terms a r e de f i ned
above. O'Conner and Dobbins p resen ted va lues o f k f rom f i e l d expe r i - 2
ments which were i n good agreement w i t h t h e i r p r e d i c t e d values.
As p a r t o f t he work p resen ted i n r e f . 2, k2 as p r e d i c t e d
by O'Connor and Dobbinsi theory was compared w i t h va lues found f o r some
o the r r i v e r s . From t h i s comparison, i t was concluded t h a t t he re was
some ques t i on about t h e v a l i d i t y o f us ing a Chezy c o e f f i c i e n t o f 17 t o
d i s t i n g u i s h between t h e two assumed types o f tu rbu lence . A lso, i t was
s t a t e d t h a t t he agreement which was found t o e x i s t between t he c a l c u l a t e d
and t he observed va lues o f k c o u l d have been cons ide rab l y b e t t e r . A 2
s i m i l a r conc lus i on was reached by Dobbins h i m s e l f (6) . A lso, OIConnor (19)
suggested us i ng Eq. 2-18 regard less o f t he va lue o f C c .
Dobbins (6) extended t he use o f Eq. 2-13 t o p r e d i c t i n g t he
r e a e r a t i o n r a t e s f o r r i v e r s and presented a genera l d i s cuss ion t o show
how r and b should be r e l a t e d t o the f l u i d p r o p e r t i e s and t he mean f l o w
parameters. Three unknown c o e f f i c i e n t s were in t roduced . Est imates
were made f o r t h e v a l u e o f two o f t h e s e c o e f f i c i e n t s , and t h e t h i r d
one was e v a l u a t e d e m p i r i c a l l y on t h e b a s i s o f t hese e s t i m a t e d va lues .
However, t h e ex t reme s c a t t e r o f t h e da ta i n t h e e v a l u a t i o n o f t h i s
c o n s t a n t d i d n o t i n s p i r e c o n f i d e n c e i n t h e a n a l y s i s p resen ted .
2.4) Comments Reaera t i on C o e f f i c i e n t s
The parameters KL and K a r e c a l l e d r e a e r a t i o n c o e f f i c i e n t s , 2
o r more p r e c i s e l y , r e a e r a t i o n r a t e c o e f f i c i e n t s . One m i g h t t e n d t o
t h i n k o f them as b e i n g dependent on t h e a b s o r p t i o n mechanism a t t h e
f r e e s u r f a c e . However, t h i s i s n o t a c t u a l l y t h e case. Cons ider t h a t
t h e w a t e r s u r f a c e i s g e n e r a l l y assumed t o be s a t u r a t e d w i t h d i s s o l v e d
oxygen. T h i s seems t o be a reasonab le assumpt ion s ince , a t t h e max-
8 imum a b s o r p t i o n r a t e s , o n l y abou t 1 o u t o f e v e r y 10 mo lecu les o f
oxygen wh ich h i t s t h e s u r f a c e goes i n t o s o l u t i o n ( 1 8 ) . I f t h e s u r f a c e
i s s a t u r a t e d , t h e a c t u a l a b s o r p t i o n p rocess i t s e l f can n o t be t h e l i m i t -
i n g f a c t o r i n d e t e r m i n i n g how f a s t t h e d i s s o l v e d oxygen c o n t e n t o f t h e
wa te r i nc reases . Net a b s o r p t i o n o f oxygen can t a k e p l a c e o n l y as f a s t
as t h e d i s o l v e d oxygen i s removed f rom t h e s u r f a c e . Thus, t h e reaera-
t i o n r a t e e f f e c t i v e l y depends on t h e downward r a t e o f t r a n s p o r t o f
d i s s o l v e d oxygen a t t h e s u r f a c e . T h i s r a t e o f t r a n s p o r t , depends i n
t u r n , on t h e t u r b u l e n c e c o n d i t i o n s near t h e f r e e s u r f a c e . I n some
o f t h e l i t e r a t u r e rev iewed above, t h e concept o f s u r f a c e renewal
was used t o r e p r e s e n t t h e i n f l u e n c e o f t u r b u l e n c e on t h i s t r a n s -
p o r t r a t e . The data p r e s e n t e d i n t h i s r e p o r t t ends t o i n d i c a t e t h a t
a d i f f u s i o n model may a l s o be used t o r e p r e s e n t t h e t r a n s p o r t r a t e .
I f t he d i f f u s i o n model i s v a l i d , one may w r i t e by analogy t o
F i c k ' s Law ( E ~ . 2-5)
where e i s t h e d i f f u s i o n c o e f f i c i e n t a t t he su r f ace and may d i f f e r f rom
Dm, i n genera l . Fur ther , i f t he f i l m i s d e f i n e d as t he reg ion i n which
a s teep concen t ra t i on g r a d i e n t e x i s t s immediately below t he f r e e sur face
and i f t h e r e i s a l i n e a r g r a d i e n t i n t h i s f i l m , then
e K = - L b 2- 20
i f Eq. 2-8 i s t a k e n a s a g e n e r a l d e f i n i t i o n f o r K T h i s e q u a t i o n g i v e s L '
KL as t he r a t i o between t he d i f f u s i o n c o e f f i c i e n t a t t he su r f ace and t he
th ickness o f t he h y p o t h e t i c a l su r f ace f i l m . The f i l m t h i c kness i s used
o n l y as a means f o r d e f i n i n g t he concen t ra t i on g r a d i e n t a t t he f r e e
sur face. Both e and b no doubt a r e r e l a t e d t o t h e tu rbu lence c o n d i t i o n s
near t he f r e e sur face. Eq. 2-20 i s presented, n o t because i t o f f e r s an
immediate means f o r p r e d i c t i n g K va lues b u t r a t h e r because i t may o f f e r L
a means f o r e v e n t u a l l y r e l a t i n g KL t o t u rbu lence and t o mean h y d r a u l i c
parameters.
Whether t h e d i f f u s i o n model o r t he su r f ace renewal model i s
used, i t must be r e a l i z e d t h a t t he mechanism which c o n t r o l s t he r a t e
o f r e a e r a t i o n i s t h e downward t r a n s p o r t o f d i s s o l v e d oxygen a t t h e f r ee
sur face, o r e f f e c t i v e l y t he tu rbu lence near t h e f r e e su r face . Thus,
seems t h a t KL concep tua l l y a b e t t e r c o e f f i c i e n t t o use t h a t K 2'
18
s i nce f o r a g iven tu rbu lence c o n d i t i o n near t he sur face K would no t L
change f o r d i f f e r e n t depths b u t K would change. Consider t he d i f - 2
f e r e n t s i g n i f i c a n c e o f K and KL. From Eq. 2-8, t he t r a n s p o r t r a t e i s 2
Y = - A K D
s L 2- 21
w h i l e f rom Eq. 1-1, cons ide r i ng t he r e l a t i o n between dD/dt and q one Y '
may a l s o w r i t e
Thus, K represents t he t r a n s p o r t ( o r absorp t ion) r a t e pe r u n i t s u r f a c e L
area w h i l e k i s a r a t e pe r u n i t volume ( v ) . I f t he mean depth h i s 2
de f i ned by h = V/As, then
so t h a t i n seeking mathematical c o r r e l a t i o n s i t makes l i t t l e d i f f e r e n c e
whether KL o r K i s used as long as h i s cons idered as one o f t he param- 2
e t e r s on which t he reae ra t i on c o e f f i c i e n t depends.
3) DIFFUSION MEASUREMENTS
As p o i n t e d o u t i n Eq. 2-20 and t h e accompanying d iscuss ion,
i t may be p o s s i b l e t o represent t he downward t r a n s p o r t o f oxygen away
from t h e f r e e su r f ace by a d i f f u s i o n model and t o r e l a t e t h e r e a e r a t i o n
c o e f f i c i e n t K t o t he d i f f u s i o n c o e f f i c i e n t i n t he su r f ace f i l m which L
i s d e f i n e d f o r p resen t purposes as t h e reg ion o f t h e steep concentra-
t i o n g r a d i e n t immediately below t he f r e e sur face.
3.1) D i f f u s i o n Equat ion
For t h e case o f a conse rva t i ve substance which i s absorbed
a t t h e f r e e su r face t u r b u l e n t water and d i f f u s e d downward i n t he absence
o f any n e t v e l o c i t i e s , t h e equa t ion f o r t he conserva t ion o f t h a t sub-
s tance may be w r i t t e n (1, p . 627-9) as
where c i s t he concen t ra t i on o f t h e substance ( t u r b u l e n t f l u c t u a t i o n s
hav ing been averaged o u t ) , y i s t h e v e r t i c a l coord ina te , e i s t h e Y
v e r t i c a l d i f f u s i v i t y f o r t he substance under c o n s i d e r a t i o n and may be
a f u n c t i o n o f y, and t i s t ime. I n genera l , i f c i s measured as a
f u n c t i o n o f y and t, t h i s data may be used t o c a l c u l a t e t h e eddy d i f -
f u s i v i t y and thereby g a i n some i n s i g h t i n t o t h e t r a n s p o r t process and
t h e way i n w h i c h i t i s a f f e c t e d by t h e t u rbu lence which i s p resen t .
3.2) Determinat ion - o f D i f f u s i o n C o e f f i c i e n t
I f a steady s t a t e c o n c e n t r a t i o n d i s t r i b u t i o n can be obta ined,
then a c h t = 0 and t h e d i f f u s i o n c o e f f i c i e n t can be c a l c u l a t e d r a t h e r
e a s i l y . I f t h e substance which i s be ing absorbed across t h e f r e e sur-
face i s e i t h e r be ing used-up a t t he bot tom o f t h e water o r i s d i f f u s i n g
o u t t h e bot tom a t t h e same r a t e t h a t a b s o r p t i o n i s t a k i n g p l a c e a t t he
sur face, then a steady s t a t e c o n c e n t r a t i o n d i s t r i b u t i o n should r e s u l t .
As descr ibed l a t e r , these exper iments sought t o use a steady s t a t e
c o n c e n t r a t i o n d i s t r i b u t i o n . For t h i s s i t u a t i o n ( i . e a c / d t = O), Eq. 3-1
can be i n t e g r a t e d once w i t h respec t t o y t o g i v e
where m i s a cons tan t . I f e and dc/dy a r e known a t some y (aga i n t he Y
s i t u a t i o n f o r these exper iments) , m can be found. The g r a d i e n t a t o t he r
va lues o f y ( i .e., dc/dy) can then be used t o eva lua te e as a f u n c t i o n Y
o f y. The cons tan t m i s p r o p o r t i o n a l t o the v e r t i c a l f l u x o f t he d i f -
f u s i n g substance. The exac t r e l a t i o n between t h e cons tan t i n Eq. 3-2
and the v e r t i c a l f l u x depends on t h e way i n which the concen t ra t i on c
i s de f ined . (see r e f . 1, Chap. 20.)
3.3) Exper imenta l Proqram
To t h e w r i t e r ' s knowledge, t h e r e i s no sensor which i s phys i -
c a l l y smal l enough t o a l l o w f o r t he measurement o f the steep d i sso l ved
oxygen g rad ien t s which appa ren t l y e x i s t i n t he f i l m reg ion . S ince smal l
temperature sensors a r e a v a i l a b l e , i t was decided t o use hea t as a
t r a c e r t o study t he d i f f u s i o n downward from a f r e e sur face. A
s i t u a t i o n analogous t o t h a t f o r oxygen abso rp t i on was c rea ted f o r
t he l a b o r a t o r y s tudy. I n a m ix i ng vessel , t he a i r over t he water
was heated t o a temperature h igher than t he water temperature. The
temperature d i f f e r e n c e caused heat t o be absorbed by t h e water a t t he
f r e e sur face and d i f f u s e d downward j u s t as w i t h oxygen i n t he reaera-
t i o n process. The s ides o f t he vessel were i n s u l a t e d w i t h 12 i n . o f
f i b e r g l a s s t o assure t h a t d i f f u s i o n o f heat took p l a c e on l y i n t he
v e r t i c a l d i r e c t i o n . Temperatures were measured i n t he water as a
f u n c t i o n o f t ime and d i s tance below the f r e e sur face. The temperature
data was used t o c a l c u l a t e then d i f f u s i o n c o e f f i c i e n t i n accordance
w i t h Eq. 3-2.
3.3.1) M ix ing Vessel
The tank i n which t he temperature p r o f i l e s were measured was
a 20" x 12" x 12" deep vessel i n which s t a i n l e s s s t e e l screens osc i l-
l a t e d v e r t i c a l l y t o produce tu rbu lence . The s i x screens ( f l a t t e n e d
expended metal , 1/2" no. 16) had a v e r t i c a l spacing o f 1" and were
d r i v e n by f ou r 5/16" rods. (see F ig . 3-1 .) The screens were fastened
w i t h nu t s and washers t o the f ou r 5/16'! threaded s t a i n l e s s s t e e l rods,
which extended through t he screen openings. The top screen had a n u t
and washer on l y on i t s lower s ide . T h i s arrangement a l lowed t he top
screen t o come c l o s e t o t he water su r face w i t h o u t hav ing a nu t on top
o f t he screen which would break t he water sur face. Each s t a i n l e s s
- r o t a t i n g disk
SQep pul l e y 5 f lywheel b e c c e n t r i c d r i v e p i n on 1 /2" s h a f t behind r o t a t i n g e c c e n t r i c i t y adjustment
d i s k
IF 1/2" x 12" rod
I------ knuckle j o i n t
?? /--mounting bracket
1*Tk4 5 / 1 6 d r i v e
- tank
screens
F ig . 3 - 1 : Schematic diagram o f m i x i n g vessel
2 2
s l o t
rods
r o d was th readed j u s t up t o t h e l e v e l o f t h e t o p screen so t h a t o n l y
t h e smooth r o d (no t h r e a d s ) ex tended th rough t h e w a t e r s u r f a c e .
Exper iments were r u n w i t h a 7/16" s t r o k e between v e r t i c a l
ex t remes o f t h e screen movement and a t 100, 150, and 200 rpm. The
i average d i s t a n c e o f t h e t o p screen be low t h e w a t e r s u r f a c e c o u l d be
changed by v a r y i n g t h e amount o f w a t e r i n t h e tank . The a i r space
o v e r t h e w a t e r i n t h e t a n k was hea ted by t h e use o f s i l i c o n e - r u b b e r -
1 embedded h e a t i n g tapes. W i t h these tap'es, t h e chance o f h e a t i n g by
r a d i a t i o n was s m a l l because o f t h e r e l a t i v e l y low tempera tu re o f t h e
tapes. Never the less , aluminum f o i l s h i e l d i n g was used t o a s s u r e t h a t
no hea t was t r a n s f e r r e d t o t h e w a t e r by r a d i a t i o n .
I ' 3.3.2) T h e r m i s t o r C i r c u i t
I The tempera tu re sensor was a 0.014"-dia., g lass -coa ted , bead
i t h e r m i s t o r ( ~ e n w a l E l e c t r o n i c s , Framingham, Massachuset ts, no. ~ ~ 3 2 ~ 1 ) .
I t was mounted a t t h e end o f a c o n i c a l nose on a 1/4" O.D. p l e x i g l a s s
tube.
As shown i n F i g . 3-2, t h e sensor was one l e g o f a Wheatstone
b r i d g e . The b r i d g e was connected t o a Sanborn r e c o r d e r (model 140) f o r
e x c i t a t i o n and a m p l i c a t i o n o f t h e o u t p u t f rom t h e b r i d g e . V o l t a g e
a c r o s s t h e sensor was r e s t r i c t e d t o l e s s than 30 m i l l i v o l t s i n o r d e r
0 t o keep t h e sensor f rom h e a t i n g more t h a n 0.01 C above i t s s u r r o u n d i n g s .
An ammeter 50 m i l l i a m p s ) was used t o i n d i c a t e when t h e b r i d g e was
ba lanced. The ammeter c i r c u i t had a c o a r s e and f i n e s w i t c h wh ich r e f e r -
r e d t o t h e s e n s i t i v i t y o f t h e c i r c u i t .
Usual o p e r a t i o n o f t h e tempera tu re measur ing d e v i c e was t o
ba lance t h e b r i d g e w i t h t h e the rm i s t o r a t a known tempera tu re u s i n g
t h e 1000 ohm p o t e n t i o m e t e r ( F i g . 3-2) and , then t o l o c k t h i s p o t e n t i o -
meter . Temperature measurement was accompl ished u s i n g t h e 100 ohm
p o t e n t i o m e t e r t o b r i n g t h e b r i d g e back t o ba lance as t h e r e s i s t a n c e
o f t h e t h e r m i s t o r changed due t o tempera tu re changes. The 100 ohm
f i x e d r e s i s t o r c o u l d be added t o t h e c i r c u i t t o g i v e a t o t a l v a r i a -
t i o n o f 200 ohms i n one l e g o f t h e b r i d g e . T h i s a l l o w e d tempera tu re
0 measurement ove r a range o f 3.7 C f rom t h e ba lance p o i n t .
The t h e r m i s t o r was c a l i b r a t e d a g a i n s t a c a l o r i m e t e r thermom-
e t e r (ASTM 56c) wh ich was marked d i r e c t l y t o 0.02'~. The c a l i b r a t i o n
c u r v e had a s l i g h t c u r v a t u r e , b u t c o u l d be c o n s i d e r e d l i n e a r o v e r a
0 range o f abou t 2 C. The s c a t t e r i n g o f t h e data on t h e c a l i b r a t i o n
c u r v e was a l l w i t h i n _ + 0 . 0 5 ~ ~ on t h e b e s t f i t l i n e ( F i g . 3 -3 ) .
3.3.3) Secondary Mechanisms -- o f Heat T r a n s f e r t o t h e Water
S ince h e a t was b e i n g used as a t r a c e r t o measure d i f f u s i o n
r a t e s , i t was necessary t o i n v e s t i g a t e p o s s i b l e hea t l o s s e s f rom t h e
m i x i n g v e s s e l and p o s s i b l e heat t r a n s f e r t o t h e w a t e r o t h e r than a c r o s s
0 t h e f r e e s u r f a c e . For t h e w a t e r 10 C warmer than room temperature , t h e
hea t l o s s th rough t h e i n s u l a t e d w a l l s was c a l c u l a t e d t o g i v e a tempera-
t u r e change o f about 0.04'C/hr. The re fo re , t h i s mechanism f o r h e a t l o s s
was c o n s i d e r e d n e g l i g i b l e , e s p e c i a l l y s i n c e t h e wa te r tempera tu re nor -
0 m a l l y was much l e s s than 10 C above room temperature . A l s o , under t y p i -
c a l o p e r a t i n g c o n d i t i o n s , b u t w i t h t h e w a t e r s u r f a c e cove red t o p r e v e n t
heat t r a n s f e r , i t was e m p i r i c a l l y de te rm ined t h a t hea t t r a n s f e r t h rough
t h e t a n k w a l l s and t h e s t e e l rods d i d n o t cause any s i g n i f i c a n t
h e a t i n g o f t h e w a t e r .
3.3.4) P rocedure
Water was added t o t h e m i x i n g v e s s e l a n d brought t o t h e de-
s i r e d t e m p e r a t u r e by immersion h e a t e r s , and t h e a i r space was heated.
The t e m p e r a t u r e o f t h e a i r space i n t h e t a n k was a u t o m a t i c a l l y con-
t r o l l e d t o w i t h i n abou t 2 0 . 2 ' ~ . The t e m p e r a t u r e p r o b e was mounted
i n a p o i n t gauge a t t h e m i d d l e o f t h e t a n k ' s l e n g t h and 3-1/4" i n
f r o m one s i d e . W i th t h e screens stopped, t h e p o i n t gage was used
t o l o c a t e t h e w a t e r s u r f a c e . The sc reens were t h e n s t a r t e d , and
measurements o f t empera tu re a t v a r i o u s dep ths and t i m e s were then
t a k e n . Repeated v e r t i c a l t e m p e r a t u r e t r a v e r s e s were made t o a s s u r e
t h a t s teady s t a t e had been reached.
I t was necessary t o c a l i b r a t e t h e t h e r m i s t o r each day t h a t
da ta was taken, b u t c a l i b r a t i o n checks a t t h e b e g i n n i n g and end o f
0 each s e t o f d a t a showed t h a t t h e t h e r m i s t o r gave t h e accu racy 2 .05 C
p r e v i o u s l y ment ioned.
3.3.5) R e s u l t s
The amount o f da ta t h a t was ob ta i n e d and t h a t c o u l d be con-
s i d e r e d r e l i a b l e was q u i t e s m a l l , due t o many e x p e r i m e n t a l p rob lems.
The g r e a t e s t p rob lem was s low, f l u c t u a t i n g t e m p e r a t u r e v a r i a t i o n s near
t h e s u r f a c e . These v a r i a t i o n s rende red a l a r g e p a r t o f t h e da ta essen-
t i a l l y u s e l e s s . I t i s supposed t h a t t h e o s c i l l a t i n g screens s e t up a
s low c i r c u l a t i o n i n t h e t a n k i n a d d i t i o n t o t h e m i x i n g t h a t wou ld be
expected from the v e r t i c a l movement o f t h e screens. Extreme ca re was
used i n t r y i n g t o assure t h a t t he screens were h o r i z o n t a l and f l a t and
t h a t t he d r i v e rods were v e r t i c a l i n o rde r t o t r y t o p reven t t he c i r -
c u l a t i o n from developing. I n s p i t e o f a l l these p recau t ions , t he c i r -
c u l a t i o n s t i l l seemed t o develop i n a r a t h e r unp red i c t ab le fash ion .
I t does no t seem t h a t the temperature f l u c t u a t i o n s c o u l d have been due
t o se iche- type waves s i nce t h e p e r i o d f o r such waves would have been
much s h o r t e r than t he p e r i o d o f t he a c t u a l f l u c t u a t i o n s (o rder o f
5 sec). Also, t h e f l u c t u a t i o n s were no t c o r r e l a t e d w i t h t he on -o f f
c y c l e o f t he hea te rs which were used t o c o n t r o l t h e a i r temperature.
Three se t s o f data t h a t d i d have an e s s e n t i a l l y steady
temperature d i s t r i b u t i o n a r e shown i n F igs . 3-4, 5, 6, a long w i t h
c o n d i t i o n s under which each se t was taken. The va lues o f the d i f -
f u s i o n c o e f f i c i e n t s a t the water su r f ace a r e a l s o shown. These va lues
were c a l c u l a t e d by Eq. 3-2, us ing t h e f a c t t h a t i n the r eg ion occupied
by t he screens (8)
where a i s t h e amp l i tude o f t h e o s c i l l a t i o n and o i s the f requency.
A l though t h i s i s c e r t a i n l y very l i m i t e d data, t h e r e a re some
p o i n t s wor th n o t i n g . For t he lowest frequency o f o s c i l l a t i o n (100 rpm,
F i g . 3-4) t h e d i f f u s i o n c o e f f i c i e n t i s approx imate ly 1.3 t imes t h e
2 mo lecu la r c o e f f i c i e n t (0.0015 cm /sec, r e f . 1 ) . When the frequency was
increased t o 200 rpm ( ~ i g . 3-5), t he d i f f u s i o n c o e f f i c i e n t increased by
a f a c t o r o f about 2.5. Th i s seems t o be a much more reasonable change
0 -. IA rr rn 3 0 m CT m 3
rr 3-
B Z'O : 1
t h a n t h e v a r i a t i o n t h a t was found i n t h e s u r f a c e renewal r a t e when t h e
f requency was inc reased . (see S e c t i o n 2.3.)
F i g . 3-6 shows data taken when t h e t o p screen came r i g h t up
t o t h e w a t e r s u r f a c e b u t d i d n o t b r e a k t h e s u r f a c e . For t h i s case,
t h e tempera tu re d i s t r i b u t i o n was s t i l l l i n e a r . There was an i n c r e a s e
by a f a c t o r o f a p p r o x i m a t e l y 75 i n t h e d i f f u s i o n c o e f f i c i e n t a s com-
p a r e d t o F i g . 3-4, b u t t h e d i f f u s i o n c o e f f i c i e n t was s t i l l abou t 240
t imes l e s s than t h a t i n t h e l ower r e g i o n s o f t h e wa te r .
A s u r f a c e f i l m t h i c k n e s s may be d e f i n e d by e x t e n d i n g t h e
two l i n e a r p o r t i o n s o f t h e tempera tu re d i s t r i b u t i o n u n t i l t hey i n t e r -
s e c t . For t h e oxygen a b s o r p t i o n proglem, t h i s f i l m t h i c k n e s s a n d t h e
d i f f u s i o n c o e f f i c i e n t a t t h e s u r f a c e wou ld be s u f f i c i e n t t o de te rm ine
t h e r e a e r a t i o n c o e f f i c i e n t K ( ~ q . 2-20). The f a c t t h a t t hese two L
s t r a i g h t l i n e s do n o t c o r r e c t l y r e p r e s e n t t h e e n t i r e c o n c e n t r a t i o n
( tempera tu re ) d i s t r i b u t i o n does n o t r e a l l y m a t t e r s i n c e o n l y t h e grad-
i e n t and t h e d i f f u s i o n c o e f f i c i e n t a t t h e s u r f a c e a r e needed t o d e t e r -
mine t h e r a t e o f a b s o r p t i o n .
3.3.6) Summary
On t h e b a s i s o f v e r y l i m i t e d data i t appears t h a t t h e r e may
be some v a l i d i t y t o t h e use o f a d i f f u s i o n model t o r e p r e s e n t t h e down-
ward t r a n s p o r t immedia te ly be low t h e f r e e s u r f a c e when a substance i s
b e i n g absorbed a t t h e f r e e s u r f a c e . The d i f f u s i o n c o e f f i c i e n t a t t h e
s u r f a c e a p p a r e n t l y can v a r y upwards f r o m t h e m o l e c u l a r v a l u e as t h e
amount o f m i x i n g a t t h e s u r f a c e i nc reases .
I f more ex tens ive t e s t s v a l i 'date the d i f f u s i o n model, then
i t would be hoped t h a t t he d i f f u s i o n c o e f f i c i e n t a t the sur face cou ld
be r e l a t e d t o the na tu re o f t he turbu lence. P a r t o f t h i s work would
be t o seek t o e x p l a i n t he mechanism f o r t he d i f f u s i o n i n v iew o f the
f a c t t h a t t he d i f f u s i o n c o e f f i c i e n t can apparen t l y l i e between t he
molecular va lue and t he t u r b u l e n t value, even when t he re i s s t rong
m ix i ng a t the sur face ( ~ i g . 3-6). The exp lana t i on f o r t h i s may be
i n t he f a c t t h a t t he substance absorbed a t t he sur face moves i n t o
the t u r b u l e n t eddies under molecular a c t i o n a t the same t ime t h a t t he
eddies a re moving about.
4. TURBULENCE MEASUREMENTS
The measurements descr ibed i n t h i s s e c t i o n were undertaken
i n an e f f o r t t o ga in some i n s i g h t i n t o t h e t u rbu lence c o n d i t i o n i n
open channel shear f l o w near the f r e e su r f ace and p o s s i b l y i n t h e f i l m
r eg i on i t s e l f . H o t - f i l m anemometry was used i n t h i s s tudy t o measure
c h a r a c t e r i s t i c s o f turbu lence. The anemometer g i v e s e s s e n t i a l l y in -
stantaneous, p o i n t v e l o c i t y readings i n t h e form o f a cont inuous out-
p u t vo l t age . Th i s cont inuous v o l t a g e s i g n a l can be used t o o b t a i n
t u rbu lence i n f o rma t i on such as the root-mean-square (RMS) and energy
spectrum o f t he v e l o c i t y f l u c t u a t i o n s .
4.1) Turbulence C h a r a c t e r i s t i c s
The p r imary c h a r a c t e r i s t i c s o f t he tu rbu lence on which data
was sought were t he i n t e n s i t y and t he one-dimensional spectrum. The
i n t e n s i t y i s de f i ned as
- where u i s t he t ime-averaged v e l o c i t y and u ' i s t he t u r b u l e n t f l u c u a t i o n ,
- i .e. u ' = u - u, where u i s t he t o t a l ins tantaneous v e l o c i t y . The one-
dimensional spectrum ~ ( n ) i s de f i ned so t h a t ~ ( n ) d n i s p r o p o r t i o n a l t o
t h e k i n e t i c energy o f t h e tu rbu lence i n t he f requency band n t o n + dn,
where n i s frequency; thus, ( l l ) ,
The n o r m a l i z e d spectrum, ~ ( n ) , i s s i m i l a r l y d e f i n e d by
so t h a t
Except f o r t h e h i g h f r e q u e n c i e s , t h e n o r m a l i z e d spectrum s h o u l d have
t h e fo rm (11, 15, 21)
u F (n) = 7
where L i s t h e macrosca le . X
4.2) Anemometry E q u i p m e n t ~ d Measurement Techniques
I n these exper iments , a c o n s t a n t t empera tu re anemometer e eat
F l u x System Model 1010 manu fac tu red by Thermo-Systems I n c o r p o r a t e d o f
S t . Pau l , ~ i n n e s o t a ) was used w i t h a Thermo-Systems h o t - f i l m sensor
h a v i n g a 0.006" d iamete r q u a r t z c o a t e d p l a t i n u m - f i l m sensor and a sens i -
t i v e l e n g t h o f 0.080". D e t a i l e d i n f o r m a t i o n on t h e p r i n c i p l e o f opera-
t i o n and techn iques f o r u s i n g a h o t - f i l m anemometer may be found i n
r e f . 11 and i n t h e m a n u f a c t u r e r ' s p u b l i c a t i o n . The sensor r e s i s t a n c e
i s r e l a t e d t o t h e o p e r a t i n g tempera tu re by t h e e q u a t i o n
i n which R i s t he r es i s t ance o f t h e sensor a t t h e ope ra t i ng tempera-
t u r e ( tS) and Re i s t he " co l d res is tance" a t t h e temperature (te) o f
t h e environment. The r a t i o R/Re i s c a l l e d the "overheat r a t i o . " The
c o e f f i c i e n t p i s t he temperature c o e f f i c i e n t f o r t h e f i l m m a t e r i a l and
i s g i ven by t he manufacturer as 0 . 0 0 2 3 4 / ~ ~ f o r t h e i r p l a t i n u m f i l m s .
An overheat r a t i o o f 1.05 was used f o r a l l t e s t s . Th i s corresponds
0 t o sensor temperature 21.4 C above t he water temperature, e.g. a
, sensor temperature o f 46.4 '~ f o r a water temperature o f 25'~.
The anemometer was c a l i b r a t e d i n t h e f lume by us ing a pre-
c a l i b r a t e d c u r r e n t meter t o determine t he v e l o c i t y a t t h e sensor. A
t y p i c a l c a l i b r a t i o n cu rve i s shown i n F ig . 4-1. The c u r r e n t meter
was a Stevens Midget Current Meter which has a p r o p e l l e r t h a t i s about
1 - i n . i n diameter. Dur ing c a l i b r a t i o n , the f l o w depth was 6 i n . and
t h e h o t - f i l m sensor and t h e cen te r o f t he p r o p e l l e r meter were p laced
about 1.5" below t he f r e e su r f ace so as t o be i n a r eg ion o f an
e s s e n t i a l l y f l a t v e l o c i t y d i s t r i b u t i o n . A t y p i c a l v e l o c i t y d i s t r i -
b u t i o n on t he v e r t i c a l c e n t e r l i n e i s shown i n F i g . 4-2.
0 The sensor was mounted on an 80 elbow on t h e bottom o f a
v e r t i c a l t r a v e r s i n g rod. T h i s elbow a l lowed the sensor t o be brought
up below t h e f r e e su r f ace w i t h o u t b reak ing t h e sur face.
The c i r c u i t r y and ins t ruments used t o process t he ou tpu t o f
t h e anemometer a r e shown schemat ica l l y i n F i g . 4-3. As w i l l be discus-
sed i n more d e t a i l l a t e r , t h e t u rbu lence had s i g n i f i c a n t components a t
f requenc ies l e s s than 2.5 Hz. These low frequency components i n t e r f e r e d
C i r c u i t A --------
_ I - - - - _
C i r c u i t B ---- ----
F i g . 4-3: Schematic diagram o f data a n a l y s i s c i r c u i t s
w i t h a t tempts t o use t h e anemometer panel-meter f o r read ing t h e D.C.
v o l t a g e which i s p r o p o r t i o n a l t o t he mean v e l o c i t y . Thus, an ex-
t e r n a l c i r c u i t (A, F ig . 4-3) was used t o f i l t e r ou t t h e low frequency
components. The R-C f i l t e r shown has a co rne r frequency o f 0.16 Hz.
C i r c u i t A, F i g . 4-3, was used t o c a l i b r a t e t h e sensor and t o measure
mean v e l o c i t i e s .
The tu rbu lence data was analyzed by use o f C i r c u i t B shown
i n F ig . 4-3. The b r i d g e v o l t a g e f rom t h e anemometer was f ed i n t o the
Sound and V i b r a t i o n Analyzer , which i s e s s e n t i a l l y a tuneable band
pass f i l t e r w i t h cen te r f requenc ies v a r i a b l e from 2.5 Hz t o 25 KHz. The
l / l 0 - o c t a v e (7%) band w i d t h was used. Th i s band w i d t h i s t he one g i ven by
t he manufacturer and i s de f i ned by t h e -3db w id th . I n ana l yz i ng t h e
tu rbu lence data, a band w i d t h o f 15% was used. T h i s w i d t h was ob ta ined
as t he w i d t h o f a r ec tangu la r response cu rve o f u n i t h e i g h t which had
t he same area as the a c t u a l response curve. The ou tpu t o f t he Sound
and V i b r a t i o n ana lyzer was t h a t p a r t o f t he tu rbu lence s i g n a l pass ing
through t he se lec ted band. T h i s ou tpu t was f ed i n t o a Hewlet t -Packard
Model H12-3400A RMS vo l tme te r modi f i e d t o have a lower frequency re-
sponse (-4db) o f 2Hz. Again, i t was found t h a t t he tu rbu lence components
a t f requenc ies l e s s than 2.5 Hz i n t e r f e r e d w i t h reading t he RMS v o l t -
meter, so t he DC ou tpu t o f t he RMS meter was f ed through another R C
f i l t e r t o an e x t e r n a l meter. The v o l t a g e read from C i r c u i t B, F ig . 4-3,
was used t o o b t a i n t he s p e c t r a l data i n accordance w i t h t he d e f i n i t i o n s
i n Sec t ion 4.1.
I t should be no ted t h a t t he use o f t h e c i r c u i t s shown inc lud-
i ng t h e RC f i l t e r s d i d n o t a l l o w measurement o f the tu rbu lence components
a t f requenc ies l e s s than 2.5 Hz, i t o n l y p reven ted these components
f rom i n t e r f e r i n g w i t h t he o the r measurements. Some i n fo rma t i on about
t he low frequency components c o u l d p robab ly have been ob ta i ned w i t h
t h i s analog- type a n a l y s i s by r eco rd i ng t he anemometer ou tpu t vo l t age
on a tape recorder and p l a y i n g i t back a t a f a s t e r speed f o r a n a l y s i s .
4.3) Flume
A c l osed system r e c i r c u l a t i n g f lume was used f o r t he turbu-
l ence measurements. A p lan-v iew ske tch o f the f lume i s shown i n F ig . 4-4,
where some o f t he dimensions a r e a l s o i nd i ca ted . The f lume had h o r i -
z o n t a l bed and a smooth (epoxy-painted wood) f l o w sur face . The f l o w
s e c t i o n was 30 f t long by 11 1 /4 i n . wide by 10 1/2 i n . deep w i t h a
stream1 ined ent rance composed o f 11- i n . l ong quadrants o f an e l l i p s e
on t h e two s ides and bottom. Jus t upstream o f the en t rance t he re were
f i v e l a y e r s o f window screen (18 by 14 mesh) spaced 3/4" apa r t . The
head and t a i l boxes were each 22 i n . w ide by 34 i n . l ong by 36 in . deep
( v e r t i c a l l y ) . The water l e v e l i n t he t a i 1 box prevented drawdown o f
the water su r f ace i n t h e f lume. The tu rbu lence measurements were made
on t h e c e n t e r l i n e o f t he f lume f i v e f e e t upstream o f t he t a i l b o x . Th i s
l o c a t i o n was v e r i f i e d t o be f a r enough upstream o f t h e t a i l b o x so as
t o a v o i d t h e e f f e c t s o f t h e separa t ion due t o f l o w i n t o t he t a i l b o x .
A lso, i t was v e r i f i e d t o be f a r enough downstream o f t he en t rance so
t h a t t he boundary l a y e r was f u l l y developed.
I n o rder t o p reven t a i r bubbles f rom forming on the sensor due
t o the heated f i l m d r i v i n g d i sso l ved gases ou t o f s o l u t i o n f rom the water,
t he water was deaerated by a method p r e v i o u s l y descr ibed (17). A lso,
the water i n the f lume system was con t i nuous l y f i l t e r e d th rough a
diatomaceous e a r t h f i l t e r ( sear Model 167.4373 f i l t e r , w i t h 15 sq f t
o f f i l t e r area, f o r p o r t a b l e swimming p o o l s ) . These methods proved
s u f f i c i e n t t o keep t h e h o t f i l m f r e e o f b o t h gas bubbles and d i r t .
4.4) Resu l t s
Data was taken f o r o n l y one f l o w c o n d i t i o n , namely a depth
o f 6" and a mean v e l o c i t y o f 0.93 f ps . The corresponding Reynold
number was 104,000 based on a l e n g t h s c a l e o f 4 t imes t h e h y d r a u l i c
r ad ius .
The tu rbu lence spec t ra (E ) f o r f requenc ies o f 2.5 Hz and
g r e a t e r a r e shown i n F ig . 4-5 f o r d i s tances o f 1.0 i n . t o 0.006 i n .
below the f r e e su r face . I t should be noted t h a t t h e diameter o f t he
ho t f i l m was 0.006 i n . and t h a t the f i l m caused sane displacement o f
t he f r e e su r f ace when i t was p l aced 0.006 i n . and 0.01 i n . below t he
f r e e su r face . Also, t he accuracy o f t h e placement o f t he f i l m a t
these two d is tances below t h e su r f ace i s c e r t a i n l y open t o ques t ion
because t h e su r f ace i t s e l f had s l i g h t r i p p l e s on i t .
I t i s es t imated t h a t a lower bound f o r t he th i ckness o f t h e
f i l m f o r oxygen abso rp t i on f o r t h i s f l o w would be on t h e o rde r o f 0.001
i n . Th i s va lue was ob ta ined by use o f Eqs. 2-20 and 2-23 w i t h e taken
-5 2 as 2x10 cm /sec (an approximate va lue f o r t h e mo lecu la r d i f f u s i v i t y )
- 5 - 1 and w i t h k2 es t imated as 3x10 sec from Eq. 2-5. I n making t h i s
c a l c u l a t i o n , Manning's n o f 0.01 was assumed t o c a l c u l a t e S . Based
on t h i s o rde r o f magnitude f o r t he f i l m th i ckness , i t i s seen t h a t some
o f t h e data was p robab ly taken near o r w i t h i n t h e f i l m . Even though i t
n = frequency i n Hz
Fig. 4-5: Energy Spectra
i s n o t p o s s i b l e t o t a k e data r i g h t a t t h e s u r f a c e , i t seems u n l i k e l y
t h a t t h e c h a r a c t e r o f t h e t u r b u l e n c e wou ld undergo any g r o s s change
i n a d i s t a n c e o f 0.01 i n . , even immed ia te l y below t h e f r e e su r face .
The presence o f s m a l l waves on t h e s u r f a c e seems i n i t s e l f t o be a
tes t imony t o t h e f a c t t h a t t u r b u l e n c e e x i s t s a t t h e f r e e s u r f a c e (26) .
A t a f requency o f 2.5 Hz, t h e s p e c t r a l d i s t r i b u t i o n has a
s l o p e wh ich i n d i c a t e s t h e presence o f s i g n i f i c a n t energy i n l o w e r
f r e q u e n c i e s ( ~ i ~ . 4-5) . W i th t h e i n s t r u m e n t s used i n these t e s t s ,
i t was n o t p o s s i b l e t o measure t h e l o w e r f requency components.
The RMS o f t h e v e l o c i t y f l u c t u a t i o n s was measured as approx-
i m a t e l y 0.021 f p s by C i r c u i t B, F i g . 4-3, f o r a l l p o s i t i o n s i n t h e
upper i n c h o f t h e w a t e r . T h i s cor responds t o an i n t e n s i t y o f 0.020
based on t h e l o c a l mean v e l o c i t y o f 1.08 f p s . As a r e s u l t o f t h e ex-
c l u s i o n o f t h e l ow f requency components f rom t h e a n a l y s i s , t h e measured
v a l u e s o f t h e RMS were no doubt t o o low. For example, R a i c h l e n (21)
measured an i n t e n s i t y o f a p p r o x i m a t e l y 0.04 a t a r e l a t i v e dep th o f
0.8 (0.77 i n . below t h e f r e e s u r f a c e ) f o r a Reynolds number o f 91,100.
The a c t u a l RMS v a l u e i n t h e upper i n c h f o r t h e p r e s e n t t e s t s
was e s t i m a t e d t o be 0.035 f p s . T h i s v a l u e was o b t a i n e d by e x t r a p o l a t -
i n g t h e s p e c t r a ( E ) t o n = 0 by a c u r v e g e o ~ n e t r i c a l l y s i m i l a r t o t h a t
f o r t h e s p e c t r a o b t a i n e d by R a i c h l e n and then i n t e g r a t i n g t h e area under
t h e c u r v e t o g e t t h e RMS f l u c t u a t i o n . T h i s i s a r a t h e r i n a c c u r a t e pro-
cedure , a t b e s t , s i n c e about 1/3 t o 1 /2 o f t h e a rea l i e s under t h e
e x t r a p o l a t e d p a r t o f t h e cu rve . T h i s a l s o i n d i c a t e s t h e need i n f c l t u re
t e s t s f o r making p r o v i s i o n f o r i n c l u d i n g f r e q u e n c i e s l e s s than 2.5 Hz i n
t h e measurements.
46
An a t t e m p t was made t o e x t r a p o l a t e t h e s p e c t r a l d i s t r i b u t i o n
by a l e a s t - s q u a r e s f i t o f t h e data t o Eq. 4-4 t o f i n d v a l u e s o f Lx and > I
~ ( 0 ) . T h i s approach l e d t o n e g a t i v e v a l u e s o f t h e i n t e g r a l s c a l e L x ,
a p p a r e n t l y i n d i c a t i n g t h a t t o o few data p o i n t s were a v a i l a b l e and/or
t h e s c a t t e r i n t h e p o i n t s was t o o g r e a t and/or Eq. 4-4 does n o t des-
i 1 tribe t h e f u n c t i o n a l v a r i a t i o n o f t h e da ta . T h i s l a t t e r reason seems
p l a u s i b l e i n v iew o f t h e compar ison between R a i c h l e n ' s data (21) and
Eq. 4-4.
Us ing t h e e s t i m a t e d RMS (0.035 f p s ) and t h e l o c a l mean v e l o c i t y
o f 1.08 f p s , t h e s p e c t r a l da ta i s shown n o r m a l i z e d i n F i g . 4-6. Data
I I f r om R a i c h l e n f o r open channe l f l o w (21) and L a u f e r (15) f o r two- ,
d imens iona l c l o s e d c o n d u i t f l o w a r e a l s o shown. I n v iew o f t h e g r o s s
i ! a p p r o x i m a t i o n s made i n t h e a n a l y s i s o f t h e da ta , t h e agreement i s some-
! what s u r p r i s i n g . I
The purpose o f making these t u r b u l e n c e measurements was t o
i n v e s t i g a t e any p o s s i b l e change i n t h e t u r b u l e n c e s t r u c t u r e i n t h e
v i c i n i t y o f t h e f r e e s u r f a c e i n v iew o f t h e s teep c o n c e n t r a t i o n grad-
i e n t s wh ich a p p a r e n t l y e x i s t i n t h e s o - c a l l e d " f i l m " . The data i n F i g . 4-5
t e n d t o i n d i c a t e t h a t t h e r e i s no s i g n i f i c a n t change i n t h e t u r b u l e n c e
s t r u c t u r e i n t h e r e g i o n immedia te ly be low t h e f r e e s u r f a c e . Thus, appar-
e n t l y , t h e s teep g r a d i e n t s and t h e l ow d i f f u s i o n r a t e s i n t h e f i l m r e g i o n
( s e c t i o n 3 ) a r e n o t caused by a decrease i n t h e t u r b u l e n c e l e v e l a t t h e
f r e e s u r f a c e . Even though data was n o t t aken on t h e p o r t i o n o f t h e
spect rum a t f r e q u e n c i e s l e s s than 2.5 Hz, r a p i d v e r t i c a l changes i n these
components wou ld c e r t a i n l y n o t be expec ted s i n c e t h e l ow f r e q u e n c i e s
r o u g h l y co r respond t o t h e l a r g e edd ies .
5) CONCLUSIONS
Reaerat ion r a t e c o e f f i c i e n t s have been r e l a t e d t o f l o w
parameters p r i m a r i l y by e m p i r i c a l and semi -emp i r i ca l approaches i n
t he pas t . As ye t , these approaches do n o t seem t o have produced
r e l a t i o n s h i p s which en joy genera l acceptance o r which a r e a b l e t o
c o r r e l a t e c o n s i s t e n t l y w i t h bo th l a b o r a t o r y and f i e l d data. Thus,
t h e r e seems t o be a need t o i n v e s t i g a t e t he bas i c mechanisms which
i n f 1 uence the reaera t i on r a t e .
I f t he water su r f ace i s a t s a t u r a t i o n concen t ra t i on , t h e
r a t e o f r eae ra t i on i s c o n t r o l l e d by the r a t e o f downward t r a n s p o r t
o f oxygen away from the f r e e sur face . Measurements, us i ng heat
( temperature) as a t r a c e r , i n d i c a t e t h a t i t may be p o s s i b l e t o repre-
sent t h e downware t r a n s p o r t by a d i f f u s i o n model. With t h i s model,
t he r a t e o f r e a e r a t i o n i s comp le te ly de f i ned i f t h e d i f f u s i o n c o e f f i -
c i e n t and t h e steepness o f t he c o n c e n t r a t i o n g r a d i e n t a r e bo th known
a t t he water su r face . L i m i t e d measurements i n a m i x i ng vesse l w i t h a
heat t r a c e r i n d i c a t e t h a t t he d i f f u s i o n c o e f f i c i e n t a t the su r face may
be equal t o o r g r e a t e r than t he mo lecu la r d i f f u s i o n c o e f f i c i e n t , b u t
t h a t i t i s cons ide rab l y l e s s than would be expected from the amount o f
m i x i ng p resen t a t the f r e e sur face. The measurements i n d i c a t e a l i n e a r
temperature ( concen t ra t i on ) g r a d i e n t immediately below t h e f r e e sur face.
Since t r a n s p o r t a t t he f r e e su r face i s appa ren t l y the c r i t i c a l
mechanism i n the r e a e r a t i o n process, i t would seem t o be b e n e f i c i a l t o
i n v e s t i g a t e t h e f l o w c o n d i t i o n s immediately below t he f r e e su r face .
Turbu lence measurements were made from 0.006 i n . t o 1 i n . below the f r e e
su r f ace i n a l a b o r a t o r y f lume w i t h 6 i n . f l o w depth and 1 f ps v e l o c i t y .
The tu rbu lence spec t ra i n d i c a t e d no change i n tu rbu lence s t r u c t u r e i n
t h i s reg ion. Thus, apparen t l y , tu rbu lence e x i s t s r i g h t up t o t he
su r f ace and i n the r eg ion o f t e n r e f e r r e d t o as the su r f ace " f i l m " .
Much more work needs t o be done on t he mechanisms which
c o n t r o l r e a e r a t i o n r a tes . Only a very l i m i t e d amount o f r e l i a b l e data
was ob ta ined du r i ng t h i s s tudy.
Thus, t he conc lus ions presented above a r e somewhat t e n t a t i v e .
More work i s needed t o i n v e s t i g a t e t he v a l i d i t y and use fu lness o f the
d i f f u s i o n model and t o seek r e l a t i o n s h i p s o f bo th t he d i f f u s i o n coef -
f i c i e n t and the concen t ra t i on g r a d i e n t t o the tu rbu lence c o n d i t i o n s
and u l t i m a t e l y t o mean f l o w parameters. I t would p robab ly be i n s t r u c -
t i v e t o make simultaneous measurements o f tu rbu lence and d i f f u s i o n near
t he f r e e su r f ace o f open channel f l ow . Th i s should a l l o w a b e t t e r
c o r r e l a t i o n o f t he d i f f u s i o n c o e f f i c i e n t t o tu rbu lence c o n d i t i o n s and
a l l o w comparison o f parameters such as t he apparent f i l m th i ckness and
t h e m ic rosca le o f the tu rbu lence .
Acknowledgements. The exper imenta l p a r t o f t h i s work was c a r r i e d ou t
by Saeed Massoudi, an undergraduate s tudent , and A. Keramati, a grad-
ua te s tuden t .
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- 2 6 . J . 0. Hinze, c o n v e r s a t i o n w i t h a u t h o r .
SYEIBOLS
s u r f a c e a rea
a m p l i t u d e o f screen m o t i o n
t h i c k n e s s o f s u r f a c e f i l m
average c o n c e n t r a t i o n i n b u l k o f wa te r
c o n c e n t r a t i o n a t a p o i n t
s a t u r a t i o n c o n c e n t r a t i o n
oxygen d e f i c i t = c - C S
m o l e c u l a r (mass) d i f f u s i o n c o e f f i c i e n t
one-d imens iona l energy spectrum
d i f f u s i o n c o e f f i c i e n t
n o r m a l i z e d one-d imens iona l energy spectrum
wa te r depth
deoxygena t i o n ra t e c o e f f i c i en t
r e a e r a t i o n r a t e c o e f f i c i e n t (pe r u n i t volume)
r e a e r a t i o n r a t e c o e f f i c i e n t (pe r u n i t s u r f a c e area)
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